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Is the following integral able to be expressed by closed form or special functions?

$$\int_{0}^{1}\log\left(\frac{\Gamma(at+b)}{\Gamma(ct+b)}\right)\frac{dt}{t}$$, for any non-negative numbers $a,b,c$. As I know Raabe's integral can be actually evaluated. (See Raabe's integral for complex argument) I originally guess that this integral can be related to Raabe's integral.

Appreciate!

Ѕᴀᴀᴅ
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Thomas
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  • The same question on MathOverflow – Robert Israel Feb 06 '18 at 08:54
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    It boils down to $\int_{0}^{1}\frac{\log\Gamma(at+b)}{t},dt$ (hence the $c$-parameter is irrelevant), or to $\int_{0}^{a}\frac{\log\Gamma(t+b)}{t},dt $, or to $\int_{0}^{a} \psi(t+b)\log(t),dt$ by integration by parts. This is related to $\text{Li}2$ via $\sum{n\geq 0}\frac{1}{(n+a)(n+b)}=\frac{\psi(a)-\psi(b)}{a-b}$ and $\int_{0}^{1} x^n \log(x),dx = -\frac{1}{(n+1)^2}$. – Jack D'Aurizio Feb 06 '18 at 11:50
  • I am wondering if $$\int_{0}^{1}\frac{\log\Gamma(at+b)}{t}dt$$ converges when we consider as $t$ approaches to $0$. – Thomas Feb 11 '18 at 03:48

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